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Properties

Label 3525.206
Modulus $3525$
Conductor $3525$
Order $230$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

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Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the Dirichlet character
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3525, base_ring=CyclotomicField(230)) M = H._module chi = DirichletCharacter(H, M([115,92,60]))
 
Copy content gp:[g,chi] = znchar(Mod(206, 3525))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3525.206");
 

Basic properties

Modulus: \(3525\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(3525\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(230\)
Copy content comment:Order
 
Copy content sage:chi.multiplicative_order()
 
Copy content gp:charorder(g,chi)
 
Copy content magma:Order(chi);
 
Real: no
Copy content comment:Whether the character is real
 
Copy content sage:chi.multiplicative_order() <= 2
 
Copy content gp:charorder(g,chi) <= 2
 
Copy content magma:Order(chi) le 2;
 
Primitive: yes
Copy content comment:If the character is primitive
 
Copy content sage:chi.is_primitive()
 
Copy content gp:#znconreyconductor(g,chi)==1
 
Copy content magma:IsPrimitive(chi);
 
Minimal: yes
Parity: odd
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 3525.bl

\(\chi_{3525}(56,\cdot)\) \(\chi_{3525}(71,\cdot)\) \(\chi_{3525}(131,\cdot)\) \(\chi_{3525}(191,\cdot)\) \(\chi_{3525}(206,\cdot)\) \(\chi_{3525}(296,\cdot)\) \(\chi_{3525}(341,\cdot)\) \(\chi_{3525}(356,\cdot)\) \(\chi_{3525}(371,\cdot)\) \(\chi_{3525}(431,\cdot)\) \(\chi_{3525}(491,\cdot)\) \(\chi_{3525}(506,\cdot)\) \(\chi_{3525}(521,\cdot)\) \(\chi_{3525}(566,\cdot)\) \(\chi_{3525}(581,\cdot)\) \(\chi_{3525}(596,\cdot)\) \(\chi_{3525}(686,\cdot)\) \(\chi_{3525}(761,\cdot)\) \(\chi_{3525}(806,\cdot)\) \(\chi_{3525}(836,\cdot)\) \(\chi_{3525}(896,\cdot)\) \(\chi_{3525}(911,\cdot)\) \(\chi_{3525}(956,\cdot)\) \(\chi_{3525}(1046,\cdot)\) \(\chi_{3525}(1061,\cdot)\) \(\chi_{3525}(1106,\cdot)\) \(\chi_{3525}(1136,\cdot)\) \(\chi_{3525}(1181,\cdot)\) \(\chi_{3525}(1196,\cdot)\) \(\chi_{3525}(1211,\cdot)\) ...

Copy content comment:Galois orbit
 
Copy content sage:chi.galois_orbit()
 
Copy content gp:order = charorder(g,chi) [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 
Copy content magma:order := Order(chi); { chi^k : k in [1..order-1] | GCD(k,order) eq 1 };
 

Related number fields

Field of values: $\Q(\zeta_{115})$
Fixed field: Number field defined by a degree 230 polynomial (not computed)

Values on generators

\((2351,1552,2026)\) → \((-1,e\left(\frac{2}{5}\right),e\left(\frac{6}{23}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(7\)\(8\)\(11\)\(13\)\(14\)\(16\)\(17\)\(19\)
\( \chi_{ 3525 }(206, a) \) \(-1\)\(1\)\(e\left(\frac{137}{230}\right)\)\(e\left(\frac{22}{115}\right)\)\(e\left(\frac{8}{23}\right)\)\(e\left(\frac{181}{230}\right)\)\(e\left(\frac{167}{230}\right)\)\(e\left(\frac{54}{115}\right)\)\(e\left(\frac{217}{230}\right)\)\(e\left(\frac{44}{115}\right)\)\(e\left(\frac{201}{230}\right)\)\(e\left(\frac{108}{115}\right)\)
Copy content comment:Value of chi at x
 
Copy content sage:chi(x)
 
Copy content gp:chareval(g,chi,x) \\\\ x integer, value in Q/Z'
 
Copy content magma:chi(x)
 
\( \chi_{ 3525 }(206,a) \;\) at \(\;a = \) e.g. 2